Belt Tension Ratio & Capstan Equation Calculator

What does this quantity describe?

At the point of impending slip, the belt tension ratio between the tight side and the slack side depends exponentially on belt friction and wrap angle: F₁/F₂ = e^(μθ), with θ in radians. Roloff/Matek writes this as m = F₁/F₂ = e^(μβ̂) and calls m the strand-force ratio; the ratio is always evaluated with the wrap angle of the small pulley, where slip starts first. The same relationship – historically Euler's, later Eytelwein's – applies to any rope, band or belt sliding over a cylinder, which is why it is also called the capstan equation.

A sailor can hold a heavily loaded line with little hand force by wrapping it several times around a capstan or cleat: every extra turn adds 2π to the wrap angle θ and multiplies the holdable tension difference. A belt drive uses exactly this belt friction effect – the pulley is the capstan, the tight side carries the load and the slack side is the hand.

Formula and variables

F1 = F2 · e^(μ·θ)

• F1 = F2 · e^(μθ)
• F2 = F1 · e^(−μθ)
• μ = ln(F1/F2) / θ
• θ = ln(F1/F2) / μ

Choose the inputs correctly

Enter two of the four quantities. Use for θ the wrap angle at the pulley where slipping would occur first – normally the smaller pulley of an open belt drive. Use for μ the friction coefficient between the belt and the pulley surface; for V-belts and ribbed belts the effective wedge friction μ′ = μ / sin(α/2) with the groove angle α applies instead of the flat-belt value.

How to use the calculator

Select the target quantity and enter the other known values with units. The result describes the limiting case at the point of impending slip; for safe operation, the actual tension difference should keep a margin below this limit. To move from the tension ratio to the drive, take the effective tension Fₜ = F₁ − F₂ and, with belt speed, the transmitted belt power.

Worked example

Given F₂ = 100 N, μ = 0.3 and θ = 180° (π radians), substitution gives F₁ = 100 · e^(0.3·π) ≈ 256.6 N, i.e. a belt tension ratio of about 2.57.

Understand the result and units

A maximum tight-side tension of 256.6 N at 100 N slack-side tension means that at most a tension difference of about 156.6 N can be transmitted under these conditions before the belt slips. This difference is the usable circumferential force; multiplied by the pulley radius it gives the maximum transmittable torque, and multiplied by belt speed the transmittable power. Roloff/Matek expresses the same result as the pull-out ratio κ = (m − 1)/m ≈ 0.61: only 61 % of the tight-side tension is available as effective tension.

Forces in newtons (N); the friction coefficient μ is dimensionless; the wrap angle may be entered in degrees or radians – the exponent itself always requires radians (θ[rad] = θ[°]·π/180).

Typical applications

The capstan equation underlies the design of flat, V- and ribbed belt drives, conveyor drives, winches and capstans, band brakes and rope clamps. In belt design it fixes, together with the wrap angle, how much effective tension a given pretension can deliver – see the drive-element calculators for the related belt and chain tools.

Assumptions, limits and common mistakes

The equation describes the static limit immediately before slip for a massless, inextensible belt, and assumes that the whole wrap arc takes part in transmitting force. In real operation only the active arc transmits the circumferential force while the belt does not slide on the rest arc (Grashof), so the full wrap angle is a design assumption rather than a measurement. At higher belt speeds the centrifugal tension Fz = A·ρ·v² reduces the contact pressure; the capstan equation then has to be applied to the strand forces reduced by Fz, i.e. (F₁ − Fz)/(F₂ − Fz) = e^(μθ).

Common mistake: Do not substitute the wrap angle in degrees into the exponent – the calculator converts automatically, but a manual calculation must use radians. Do not use the flat-belt friction coefficient for a V-belt, and do not equate μ with the static friction of a dry, clean surface if the belt is worn, dusty or oily.